Let T:[0,1]→R be the classical Takagi function, T(x)=∑_k=0^∞2^-koperatornamedist(2^k x,Z). For y∈[0,tfrac23], define L(y)=x∈[0,1]:T(x)=y and, for each positive integer m, define S_m=y∈[0,tfrac23]:|L(y)|=m. The conjecture asserts that for every positive integer n, the set S_2n has positive Lebesgue measure: lambda(S_2n)>0.

Allaart’s conjecture on Takagi-function level sets

Allaart’s conjecture asserts that, for every positive integer $n$ , the heights with exactly $2n$ points in their Takagi-function level set have positive Lebesgue measure.

Known results

The 2011 paper states the conjecture for $S_{2n}=\{y\in[0,\tfrac{2}{3}]:|L(y)|=2n\}$ .
It proves positivity for cardinalities associated with powers of $2$ , sums of two distinct powers of $2$ , and differences of two distinct powers of $2$ .
Every positive even cardinality occurs for uncountably many level sets, but each $S_{2n}$ is nowhere dense.

September 28, 2026 claimed proof

A report identified Lai Jiang’s paper Positive measure and level sets of the Takagi function as proving that $S_{2n}$ has positive measure for every positive integer $n$ , which would settle Allaart’s conjecture. The supplied arXiv search did not independently substantiate that the paper proves the full claim, so this remains unverified; the result does not classify all level-set heights.

Current status (as of September 2026): Allaart’s abundance conjecture has a claimed complete proof, but it is unverified; classification of all level-set heights remains unsettled.

Allaart’s conjecture on Takagi-function level sets

Primary source

Additional references

Progress summary

Solutions 0